Approximating Sumset Size

TL;DR

This paper introduces a sublinear-time algorithm for estimating the size of sumsets in the Boolean hypercube, which is efficient and independent of the ambient dimension, unlike previous methods.

Contribution

It presents the first sublinear-time algorithm for sumset size estimation that works with oracle access and provides approximate results with high probability.

Findings

Algorithm's query complexity depends only on psilon, not on dimension n.

Provides psilon-approximate sumset volume with high probability.

Achieves efficiency through a novel approximation method.

Abstract

Given a subset $A$ of the $n$-dimensional Boolean hypercube $\mathbb{F}_2^n$, the sumset $A+A$ is the set $\{a+a': a, a' \in A\}$ where addition is in $\mathbb{F}_2^n$. Sumsets play an important role in additive combinatorics, where they feature in many central results of the field. The main result of this paper is a sublinear-time algorithm for the problem of sumset size estimation. In more detail, our algorithm is given oracle access to (the indicator function of) an arbitrary $A \subseteq \mathbb{F}_2^n$ and an accuracy parameter $\epsilon > 0$, and with high probability it outputs a value $0 \leq v \leq 1$ that is $\pm \epsilon$-close to $\mathrm{Vol}(A' + A')$ for some perturbation $A' \subseteq A$ of $A$ satisfying $\mathrm{Vol}(A \setminus A') \leq \epsilon.$ It is easy to see that without the relaxation of dealing with $A'$ rather than $A$, any algorithm for estimating $\mathrm{Vol}(A+A)$ to any nontrivial accuracy must make $2^{\Omega(n)}$ queries. In contrast, we give an algorithm whose query complexity depends only on $\epsilon$ and is completely independent of the ambient dimension $n$.